Theory of reproducing systems on locally compact abelian groups
نویسندگان
چکیده
A reproducing system is a countable collection of functions {φj : j ∈ J } such that a general function f can be decomposed as f = ∑ j∈J cj(f) φj , with some control on the analyzing coefficients cj(f). Several such systems have been introduced very successfully in mathematics and its applications. We present a unified viewpoint to the study of reproducing systems on locally compact abelian groups G. This approach gives a novel characterization of the Parseval frame generators for a very general class of reproducing systems on L2(G). As an application of this result, we obtain a new characterization of Parseval frame generators for Gabor and affine systems on L2(G).
منابع مشابه
On continuous cohomology of locally compact Abelian groups and bilinear maps
Let $A$ be an abelian topological group and $B$ a trivial topological $A$-module. In this paper we define the second bilinear cohomology with a trivial coefficient. We show that every abelian group can be embedded in a central extension of abelian groups with bilinear cocycle. Also we show that in the category of locally compact abelian groups a central extension with a continuous section can b...
متن کاملBracket Products on Locally Compact Abelian Groups
We define a new function-valued inner product on L2(G), called ?-bracket product, where G is a locally compact abelian group and ? is a topological isomorphism on G. We investigate the notion of ?-orthogonality, Bessel's Inequality and ?-orthonormal bases with respect to this inner product on L2(G).
متن کاملShift Invariant Spaces and Shift Preserving Operators on Locally Compact Abelian Groups
We investigate shift invariant subspaces of $L^2(G)$, where $G$ is a locally compact abelian group. We show that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a Parseval frame. For a second countable locally compact abelian group $G$ we prove a useful Hilbert space isomorphism, introduce range funct...
متن کاملPseudoframe multiresolution structure on abelian locally compact groups
Let $G$ be a locally compact abelian group. The concept of a generalized multiresolution structure (GMS) in $L^2(G)$ is discussed which is a generalization of GMS in $L^2(mathbb{R})$. Basically a GMS in $L^2(G)$ consists of an increasing sequence of closed subspaces of $L^2(G)$ and a pseudoframe of translation type at each level. Also, the construction of affine frames for $L^2(G)$ bas...
متن کاملOn component extensions locally compact abelian groups
Let $pounds$ be the category of locally compact abelian groups and $A,Cin pounds$. In this paper, we define component extensions of $A$ by $C$ and show that the set of all component extensions of $A$ by $C$ forms a subgroup of $Ext(C,A)$ whenever $A$ is a connected group. We establish conditions under which the component extensions split and determine LCA groups which are component projective. ...
متن کامل